Almost Periodic Function - Definition and Properties

Definition and Properties

There are several inequivalent definitions of almost periodic functions. An almost periodic function is a complex-valued function of a real variable that has the properties expected of a function on a phase space describing the time evolution of such a system. There have in fact been a number of definitions given, beginning with that of Harald Bohr. His interest was initially in finite Dirichlet series. In fact by truncating the series for the Riemann zeta function ζ(s) to make it finite, one gets finite sums of terms of the type

with s written as (σ + it) – the sum of its real part σ and imaginary part it. Fixing σ, so restricting attention to a single vertical line in the complex plane, we can see this also as

Taking a finite sum of such terms avoids difficulties of analytic continuation to the region σ < 1. Here the 'frequencies' log n will not all be commensurable (they are as linearly independent over the rational numbers as the integers n are multiplicatively independent – which comes down to their prime factorizations).

With this initial motivation to consider types of trigonometric polynomial with independent frequencies, mathematical analysis was applied to discuss the closure of this set of basic functions, in various norms.

The theory was developed using other norms by Besicovitch, Stepanov, Weyl, von Neumann, Turing, Bochner and others in the 1920s and 1930s.